Robust-method-to-retrieve-the-constitutive-effecti

RobustmethodtoretrievetheconstitutiveeffectiveparametersofmetamaterialsXudongChen,TomaszM.Grzegorczyk,Bae-IanWu,JoePacheco,Jr.,andJinAuKongResearchLaboratoryofElectronics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA(Received25February2004;published26July2004)Weproposeanimprovedmethodtoretrievetheeffectiveconstitutiveparameters(permittivityandperme-ability)ofaslabofmetamaterialfromthemeasurementofSparameters.Improvementsoverexistingmethodsincludethedeterminationofthefirstboundaryandthethicknessoftheeffectiveslab,theselectionofthecorrectsignofeffectiveimpedance,andamathematicalmethodtochoosethecorrectbranchoftherealpartoftherefractiveindex.ThesensitivityoftheeffectiveconstitutiveparameterstotheaccuracyoftheSparametersisalsodiscussed.Themethodhasbeenappliedtovariousmetamaterialsandthesuccessfulretrievalresultsproveitseffectivenessandrobustness.DOI:10.1103/PhysRevE.70.016608PACSnumber(s):41.20.Jb,42.25.Bs,78.20.CiI.INTRODUCTIONLeft-handed(LH)structureshavebeenrealizedsofarasmetamaterials[1–3]andveryquickly,researchershavebeenworkingonretrievingtheireffectivepermittivityandperme-abilitytobettercharacterizethem[4–6].Knownmethodstodate[7,8]useSparameterscalculatedfromawaveincidentnormallyonaslabofmetamaterial,fromwhichtheeffectiverefractiveindexnandimpedancezarefirstobtained.Thepermittivityeandpermeabilitymarethendirectlycalculatedfromm=nzande=n/z.Itisalsoknownthatthisretrievalprocessmayfailinsomeinstances,suchaswhenthethicknessoftheeffectiveslab(exhibitingbulkproperties)isnotaccuratelyestimated[4]orwhenreflectionsS11dandtransmissionsS21ddataareverysmallinmagnitude[6].Althoughtheseissueshavebeenaddressedtosomeextentinpreviousworks,wehavefoundthattheretrievedresultsinsomecasesarestillunsatisfac-tory.Inparticular,thedeterminationofthefirstboundaryoftheeffectivehomogeneousslab,theselectionofthesignoftherootswhensolvingfortheimpedancez,thedetermina-tionofthebranchoftherealpartofrefractiveindexn,andtheoriginofthespikesappearingintheretrievedimpedancez,aremanyproblemsthatdeservefurtherinvestigation.Theaforementionedissuesareaddressedinthenextsectionsofthispaperandsometypicalretrievalresultsarepresentedtoshowtherobustnessandeffectivenessofthemethod.Notethatthevaluesofe,m,andzarerelativetothoseinfreespace,thusdimensionless.II.RETRIEVALMETHODA.TheoreticalretrievalequationsInordertoretrievetheeffectivepermittivityandperme-abilityofaslabofmetamaterial,weneedtocharacterizeitasaneffectivehomogeneousslab.Inthiscase,wecanretrievethepermittivityandpermeabilityfromthereflectionsS11dandtransmissionsS21ddata.Foraplanewaveincidentnor-mallyonahomogeneousslabofthicknessdwiththeorigincoincidingwiththefirstfaceoftheslab,S11isequaltothereflectioncoefficient,andS21isrelatedtothetransmissioncoefficientTbyS21=Teik0d,wherek0denotesthewavenum-beroftheincidentwaveinfreespace.TheSparametersarerelatedtorefractiveindexnandimpedancezby[7,9,10]:S11=R01s1−ei2nk0dd1−R012ei2nk0d,s1adS21=s1−R012deink0d1−R012ei2nk0d,s1bdwhereR01=z−1/z+1.Asithasbeenpointedoutin[4,5],therefractiveindexnandtheimpedancezareobtainedbyinvertingEqs.(1a)and(1b),yieldingz=±Îs1+S11d2−S212s1−S11d2−S212,s2adeink0d=X±iÎ1−X2,s2bdwhereX=1/2S21s1−S112+S212d.Sincethemetamaterialun-derconsiderationisapassivemedium,thesignsinEqs.(2a)and(2b)aredeterminedbytherequirementz8ù0,s3adn9ù0,s3bdwheres·d8ands·d9denotetherealpartandimaginarypartoperators,respectively.ThevalueofrefractiveindexncanbedeterminedfromEq.(2b)asn=1k0dhfflnseink0ddg9+2mpg−iflnseink0ddg8j,s4dwheremisanintegerrelatedtothebranchindexofn8.Asmentionedabove,theimaginarypartofnisuniquelydeter-mined,buttherealpartiscomplicatedbythebranchesofthelogarithmfunction.Equations(2a)and(2b)canbedirectlyappliedinthecaseofahomogeneousslabforwhichtheboundariesoftheslabarewelldefinedandtheSparametersareaccuratelyknown.However,sinceametamaterialitselfisnothomogeneous,thetwoapparentlystraightforwardissuesmentionedaboveneedtobecarefullyaddressed.First,thelocationofthetwoPHYSICALREVIEWE70,016608(2004)1539-3755/2004/70(1)/016608(7)/$22.50©2004TheAmericanPhysicalSociety70016608-1boundariesoftheeffectiveslabneedtobedetermined,whichwedoherebyensuringaconstantimpedanceforvariousslabthicknesses.Second,theSparametersobtainedfromnumericalcomputationormeasurementsarenoisywhichcancausetheretrievalmethodtofail,especiallyatthosefrequen-cieswherezandnaresensitivetosmallvariationsofS11andS21.Thesetwoproblemsareexaminedindetailinthefol-lowingsections.B.DeterminationofthefirstboundaryandthethicknessoftheeffectiveslabAhomogeneousslabofmaterialcanbecharacterizedbythefactthatitsimpedancedoesnotdependonitsthickness.Ourunderstandingofthephysicalmeaningofthefirsteffec-tiveboundaryisaplanebeyondwhichthereflectedwavebehaveslikeaplanewaveforaplanewaveincidence.Whenaplanewaveisincidentonametamaterial,currentswillbeinducedonthemetalscreatingascatteredfield.Thefieldproducedbytheinducedcurrentsisnotuniform:Itisstron-gestaroundthemetalanddecayatacertaindistance.Thefirsteffectiveboundaryislocatedwherethereflectedwavebehaveslikeaplanewave,andithastobedetermined.Weusez1andz2to